The short proof is parallel to Bott's argument in the case of a compact Lie group
(cf.  [1]}). Indeed, by the analogue of Bott's Decomposition Lemma,
the \(E_1\) -term of the spectral sequence for
the column filtration of the truncated complex \((\Omega _{\rm alg}^{*,*}(NG)_n, \delta , d)\)  is
seen to be
\( E_1^{p,q} = H_\delta ^p \bigl (\Omega _{\rm alg}^q(G^p)_n\bigr ) \cong H_{\rm alg}^{p-q}\bigl (G; S^q(\mathfrak {g}^*)_n\bigr ) .\) 

The algebraic counterpart of the van Est isomorphism, which is a consequence of
Hochschild's isomorphism [2]}, then implies:
\( \begin{split}H_{\rm alg}^{0}\bigl (G; S^q(\mathfrak {g}^*)_n\bigr ) &\cong S^q(\mathfrak {g}^*)_n^G, \quad \hbox{and} \\H_{\rm alg}^{i}\bigl (G; S^q(\mathfrak {g}^*)_n\bigr ) &= 0 , \qquad \hbox{if} \quad i>0 .\end{split}\) 

From (REF ) and (REF ) it follows that
\(H^*(\Omega _{\rm alg}^*(NG)_n\bigr )\cong S^*(\mathfrak {g}^*)_n=P_n [c_1, c_2, \ldots , c_n] .\) 

The left hand side is the same as \(HH^*\left(\left(\Omega _{\rm alg}^*G)_n\right)\right)\) 
while the right hand is the same as \(H^*\bigl (W(\mathfrak {g}, G)_n\bigr )\) .
The Shulman-Stasheff argument in §REF  can also be adapted. The
algebraic counterpart of \(\psi _m\)  in Theorem REF  is \(\psi _m^{\rm alg} = \mathop {\rm Id}\nolimits \) ,
and the proof that
\(\varphi _m^{\rm alg}: W_1(\mathfrak {g}, G)/{\mathcal {F}}_1^{m+1} \rightarrow \Omega _{\rm alg}^*(\bar{N}G)_G/{\mathcal {F}}^{m+1} \equiv \Omega _{\rm alg}^*(NG)_m\) 
induces an isomorphism of spectral sequences follows along the same lines as in [3]},
with the difference that the semi-simplicial generalization of the van Est bicomplex [4]}
is replaced by its algebraic counterpart, which can be handled as in the proof of
[5]} for the algebraic version of the van Est isomorphism.
We finally recall that () automatically follows from (REF ).
